Please ensure that you are allowing Firefox to save cache and cookies.
This should make sure that you don't get logged out each time.

The way "Remember me" works is to save a persistent cookie (a cookie with a distant future expiration date). By default, Firefox saves cookies the way sites specify, but you can modify this in a number of ways that may result in losing that cookie:

• Browse in a private window (cookies deleted when you end the session)
• Change your cookie settings in the Privacy Options ("Firefox will: Never remember history" or "Firefox will: Use custom settings for history" with cookies set to something other than "Keep until: they expire")
• Use the feature to clear private data at shutdown (and include cookies in what gets cleared)
• Use an add-on that modifies how cookies are saved
• Use an external security, privacy, or clean-up program that removes browser passwords cookies

Details like websites remembering you (log you in automatically) are stored in a cookie.

• Create a cookie 'allow' exception to keep such cookies, especially in case of secure websites and when cookies expire when Firefox is closed.
• Tools > Options > Privacy > Cookies: Exceptions

In case you are using "Clear history when Firefox closes":

• do not clear the Cookies
• do not clear the Site Preferences

• Tools > Options > Privacy > Firefox will: "Use custom settings for history": [X] "Clear history when Firefox closes" > Settings
• https://support.mozilla.org/kb/Clear+Recent+History

Note that clearing "Site Preferences" clears all exceptions for cookies, images, pop-up windows, software installation, passwords, and other website specific data.

Clearing cookies will remove all specified (selected) cookies including cookies with an allow exception that you would like to keep.